Verified answer

divF = d/dx of (9xa^2 + 7ay^2) + d/dy of (7z^3 − 6ay) + d/dz of [−(3z + 7x^2 + 7y^2)]

= 9a^2 - 6a - 3.

a) divF = 9a^2 - 6a - 3 = 0

3(3a^2 - 2a - 1) = 0

3a^2 - 2a - 1 = 0

3a^2 + a - 3a - 1 = 0

a(3a + 1) - (3a + 1) = 0

(a - 1)(3a + 1) = 0

a = 1 or a = -1/3.

b) Since divF is a quadratic function of a with positive coefficient on a^2, it follows from the symmetry of a parabola that the value of a that minimizes divF is just the average of the zeros of divF.

So a = (-1/3 + 1)/2 = 1/3 minimizes divF.

Lord bless you today!